Precalculus (10th Edition)

Published by Pearson
ISBN 10: 0-32197-907-9
ISBN 13: 978-0-32197-907-0

Chapter 4 - Polynomial and Rational Functions - 4.6 Complex Zeros; Fundamental Theorem of Algebra - 4.6 Assess Your Understanding - Page 241: 46

Answer

The last zero must be real otherwise the polynomial function will have a degree of $5$. One of the missing zeros is $4+i$.

Work Step by Step

The Conjugate Pairs Theorem says that if a polynomial has real coefficients, then any complex zeros occur in conjugate pairs. That is, if $a + bi$ is a zero then so is $a – bi$ and vice-versa. We have that $-3$ and $(4-i)$ are zeros of $f(x)$, hence according to the Conjugate Pair Theorem $\overline{4-i}=4+i$ is also a zero of $f(x)$, hence $f(x)$ has at least 3 zeros, so in order for it to have 4 zeros, the fourth zero has to be a real (non-complex number), because if it was a complex number its conjugate pair would also be a zero, making the degree at least 5 according to the Conjugate Pair Theorem.
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